[SOLVED] EMATM0044 -Worksheet Week 21

Description

Problems

Q1. X and Y are two (binary) random variables. If X and Y are independent, then P(X,Y ) = P(X)P(Y )

• Give an example of two random variables that are independent.
• Complete the probability table below in such way that the variables X and Y are independent.

• Determine the missing entries (a, b) of the joint distribution in such a way that the variables X and Y are again independent.

P(Y = 0,X = 0) = 0.1

P(Y = 0,X = 1) = 0.3

P(Y = 1,X = 0) = a

P(Y = 1,X = 1) = b

Q2. Consider the following Bayesian network:

2

• Which random variables are independent of X_3,1?
• Which random variables are independent of X_3,1 given X_1,1?

Q3. Solve the questions on slides 42 and 44 of the lecture slides.

Q4. A patient can have a symptom, S, that is caused by two different diseases, A and B. It is known that the presence of a gene G is important in the manifestation of disease A.

The Bayes net and conditional probability tables are shown in Figure 2.

Figure 1: Bayes net and probability tables for Q5

• What is the probability that a patient has disease A

3                                               turn over/Qu. continues …

• What is the probability that a patient has disease A if we know that the patient has disease B
• What is the probability that a patient has disease A if we know that the patient has disease B AND symptom S